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If\, $a_1,\,a_2,\,\ldots,\,a_n$\, are positive numbers, we define their \emph{harmonic mean} as the inverse number of the arithmetic mean of their inverse numbers:
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If $a_1,a_2,\ldots,a_n$ are positive numbers, we define their \emph{harmonic mean} as:
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| $$H.M.=\frac{n}{\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}}$$ |
$$H.M.=\frac{n}{\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}}$$ |
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| \begin{itemize} |
If you travel from city $A$ to city $B$ at $x$ miles per hour, and then you travel back at $y$ miles per hour. What was the average velocity for the whole trip?\\ |
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The harmomic mean of $x$ and $y$!. That is, the average velocity is |
| \item If you travel from city $A$ to city $B$ at $x$ miles per hour, and then you travel back at $y$ miles per hour.\, What was the average velocity for the whole trip?\\ |
$$\frac{2}{\frac{1}{x}+\frac{1}{y}}=\frac{2xy}{x+y}$$ |
| The harmonic mean of $x$ and $y$!.\, That is, the average velocity is |
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| $$\frac{2}{\frac{1}{x}+\frac{1}{y}}=\frac{2xy}{x+y}.$$ |
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| \item If one draws through the intersecting point of the diagonals of a trapezoid a line parallel to the parallel sides of the trapezoid, then the segment of the line inside the trapezoid is equal to the harmonic mean of the parallel sides. |
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| \item In the harmonic series |
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| $$1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...$$ |
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| every \PMlinkescapetext{term equals to the harmonic mean of the term} preceding it and the term following it. |
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| \end{itemize} |
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